Variational Bayes Estimation of Discrete-Margined Copula Models with Application to Time Series

We propose a new variational Bayes estimator for high-dimensional copulas with discrete, or a combination of discrete and continuous, margins. The method is based on a variational approximation to a tractable augmented posterior, and is faster than previous likelihood-based approaches. We use it to estimate drawable vine copulas for univariate and multivariate Markov ordinal and mixed time series. These have dimension $rT$, where $T$ is the number of observations and $r$ is the number of series, and are difficult to estimate using previous methods. The vine pair-copulas are carefully selected to allow for heteroskedasticity, which is a feature of most ordinal time series data. When combined with flexible margins, the resulting time series models also allow for other common features of ordinal data, such as zero inflation, multiple modes and under- or over-dispersion. Using six example series, we illustrate both the flexibility of the time series copula models, and the efficacy of the variational Bayes estimator for copulas of up to 792 dimensions and 60 parameters. This far exceeds the size and complexity of copula models for discrete data that can be estimated using previous methods

Copulas as High Dimensional Generative Models: Vine Copula Autoencoders

We introduce the vine copula autoencoder (VCAE), a flexible generative model for high-dimensional distributions built in a straightforward three-step procedure. First, an autoencoder (AE) compresses the data into a lower dimensional representation. Second, the multivariate distribution of the encoded data is estimated with vine copulas. Third, a generative model is obtained by combining the estimated distribution with the decoder part of the AE. As such, the proposed approach can transform any already trained AE into a flexible generative model at a low computational cost. This is an advantage over existing generative models such as adversarial networks and variational AEs which can be difficult to train and can impose strong assumptions on the latent space. Experiments on MNIST, Street View House Numbers and Large-Scale CelebFaces Attributes datasets show that VCAEs can achieve competitive results to standard baselines

A study of dependency features of spike trains through copulas

Simultaneous recordings from many neurons hide important information and the connections characterizing the network remain generally undiscovered despite the progresses of statistical and machine learning techniques. Discerning the presence of direct links between neuron from data is still a not completely solved problem. To enlarge the number of tools for detecting the underlying network structure, we propose here the use of copulas, pursuing on a research direction we started in [1]. Here, we adapt their use to distinguish different types of connections on a very simple network. Our proposal consists in choosing suitable random intervals in pairs of spike trains determining the shapes of their copulas. We show that this approach allows to detect different types of dependencies. We illustrate the features of the proposed method on synthetic data from suitably connected networks of two or three formal neurons directly connected or influenced by the surrounding network. We show how a smart choice of pairs of random times together with the use of empirical copulas allows to discern between direct and un-direct interactions

A CFD-informed model for subchannel resolution crud prediction

A physics-directed, statistically based, surrogate model of the small scale flow fea- tures that impact Chalk River unidentified deposit (crud) growth is presented in this work. The objective of the surrogate is to provide additional details of the rod surface temperature, heat flux, and near-wall turbulent kinetic energy fields which cannot be explicitly captured by a subchannel code. Operating as a mapping from the high fidelity computational fluid dynamics (CFD) data to the low fidelity subchannel grid (hi2lo), the model provides CFD-informed bound- ary conditions to the crud model executed on the subchannel pin surface mesh. The surface temperature, heat flux, and turbulent kinetic energy, henceforth referred to as the fields of interest (FOI), govern the growth rate of crud on the surface of the rod and the precipitation of boron in the porous crud layer. Therefore the model predicts the behavior of the FOIs as a function of position in the core and local thermal-hydraulic (TH) conditions. The subchannel code produces an estimate for all crud-relevant TH quantities at a coarse spatial resolution everywhere in the core and executes substantially faster than CFD. In the hi2lo approach, the solution provided by the subchannel code is augmented by a predicted stochastic component of the FOI informed by CFD results to provide a more detailed description of the target FOIs than subchannel can provide alone. To this end, a novel method based on the marriage of copula and gradient boosting techniques is proposed. This methodology forgoes a spatial interpolation procedure for a statistically driven approach, which predicts the fractional area of a rod’s surface in excess of some critical temperature but not precisely where such maxima occur on the rod surface. The resultant model retains the ability to account for the presence of hot and cold spots on the rod surface induced by turbulent flow downstream of spacer grids when producing crud estimates. Sklar’s theorem is leveraged to decompose multivariate probability densities of the FOI into independent copula and marginal models. The free parameters within the copula model are predicted using a combination of supervised regression and classification machine learning techniques with training data sets supplied by a suite of precomputed CFD results spanning a typical pressurized water reactor TH envelope. Results show that compared to the subchannel standalone case, the hi2lo method more accurately preserves the influence of spacer grids on the crud growth rate. Or more precisely, the hi2lo method recovers key statistical properties of the FOI which impact crud growth. Compared to gold standard high fidelity CFD/crud coupled results in a single assembly test case, the hi2lo model produced a relative total crud mass difference of -8.9% compared to the standalone subchannel relative crud mass difference of 192.1%.Mechanical Engineerin

Probabilistic models for data efficient reinforcement learning

Trial-and-error based reinforcement learning (RL) has seen rapid advancements in recent times, especially with the advent of deep neural networks. However, the standard deep learning methods often overlook the progress made in control theory by treating systems as black-box. We propose a model-based RL framework based on probabilistic Model Predictive Control (MPC). In particular, we propose to learn a probabilistic transition model using Gaussian Processes (GPs) to incorporate model uncertainty into long-term predictions, thereby, reducing the impact of model errors. We provide theoretical guarantees for first-order optimality in the GP-based transition models with deterministic approximate inference for long-term planning. We demonstrate that our approach not only achieves the state-of-the-art data efficiency, but also is a principled way for RL in constrained environments. When the true state of the dynamical system cannot be fully observed the standard model based methods cannot be directly applied. For these systems an additional step of state estimation is needed. We propose distributed message passing for state estimation in non-linear dynamical systems. In particular, we propose to use expectation propagation (EP) to iteratively refine the state estimate, i.e., the Gaussian posterior distribution on the latent state. We show two things: (a) Classical Rauch-Tung-Striebel (RTS) smoothers, such as the extended Kalman smoother (EKS) or the unscented Kalman smoother (UKS), are special cases of our message passing scheme; (b) running the message passing scheme more than once can lead to significant improvements over the classical RTS smoothers. We show the explicit connection between message passing with EP and well-known RTS smoothers and provide a practical implementation of the suggested algorithm. Furthermore, we address convergence issues of EP by generalising this framework to damped updates and the consideration of general -divergences. Probabilistic models can also be used to generate synthetic data. In model based RL we use ’synthetic’ data as a proxy to real environments and in order to achieve high data efficiency. The ability to generate high-fidelity synthetic data is crucial when available (real) data is limited as in RL or where privacy and data protection standards allow only for limited use of the given data, e.g., in medical and financial data-sets. Current state-of-the-art methods for synthetic data generation are based on generative models, such as Generative Adversarial Networks (GANs). Even though GANs have achieved remarkable results in synthetic data generation, they are often challenging to interpret. Furthermore, GAN-based methods can suffer when used with mixed real and categorical variables. Moreover, the loss function (discriminator loss) design itself is problem specific, i.e., the generative model may not be useful for tasks it was not explicitly trained for. In this paper, we propose to use a probabilistic model as a synthetic data generator. Learning the probabilistic model for the data is equivalent to estimating the density of the data. Based on the copula theory, we divide the density estimation task into two parts, i.e., estimating univariate marginals and estimating the multivariate copula density over the univariate marginals. We use normalising flows to learn both the copula density and univariate marginals. We benchmark our method on both simulated and real data-sets in terms of density estimation as well as the ability to generate high-fidelity synthetic data.Open Acces

Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships

One of the main goals of current systems neuroscience is to understand how neuronal populations integrate sensory information to inform behavior. However, estimating stimulus or behavioral information that is encoded in high-dimensional neuronal populations is challenging. We propose a method based on parametric copulas which allows modeling joint distributions of neuronal and behavioral variables characterized by different statistics and timescales. To account for temporal or spatial changes in dependencies between variables, we model varying copula parameters by means of Gaussian Processes (GP). We validate the resulting Copula-GP framework on synthetic data and on neuronal and behavioral recordings obtained in awake mice. We show that the use of a parametric description of the high-dimensional dependence structure in our method provides better accuracy in mutual information estimation in higher dimensions compared to other non-parametric methods. Moreover, by quantifying the redundancy between neuronal and behavioral variables, our model exposed the location of the reward zone in an unsupervised manner (i.e., without using any explicit cues about the task structure). These results demonstrate that the Copula-GP framework is particularly useful for the analysis of complex multidimensional relationships between neuronal, sensory and behavioral variables